# Magnetic Field of Two Parallel Wires

#### BlackMelon

Joined Mar 19, 2015
133
Dear All,

I have a question about the magnetic field of two parallel wires. Here, we got two currents coming out of the page, and the magnetic fluxes flow just like the blue arrow. Usually, in textbooks, they recommend us to find the magnetic field resulting from each current flowing alone first using Ampere's Law, then add them up like vectors co-ordinate-by-co-ordinate.
However, if I let the two currents flow simultaneously and then choose the Amperian loop like the red circle around the current on the left, would the closed-loop integration of the magnetic field along this path equal to this current? If so, what co-ordinate system should I use? Where should I position my origin point?

Best Regards,
BlackMelon

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#### djducar

Joined Aug 5, 2022
6
The closed loop integration of amperes law will tell you the current density (i.e. current for ideal wire) within that closed loop. This is true for any arbitrary system.

In this case, whether you let only 1 current flow or 2, it should not affect the final answer of the current in the red boundary. The only difference is when they both flow, the left loses field strength to the right on the bottom, but gains some strength from the right on the top. How much they "gain" or "lose" will be directly related to the current ratio, and, if currents are equal, they will gain and lose the same amount. Doing them both independently and adding them is just a simplified method of doing this.

The coordinate system is a little more difficult, but probably polar if planning on a red circle around 1 wire. The actual difficult part will be how to coordinately shift the second wire so you can add B fields on one plane.